Helpppp me i dont understand this and i need the answer fast <3

Mathematics

1 answer:

NikAS [45]3 years ago

5 0

Answer:

it would be -4.5

Step-by-step explanation:

cause thats the middle

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Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 7x2 + xy + 7y2 = 15, (1, 1

riadik2000 [5.3K]

Assume $y=y(x)$ .

$7x^2+xy+7y^2=15$
$\dfrac{\mathrm d}{\mathrm dx}[7x^2+xy+7y^2]=\dfrac{\mathrm d}{\mathrm dx}[15]$
$14x+y+x\dfrac{\mathrm dy}{\mathrm dx}+14y\dfrac{\mathrm dy}{\mathrm dx}=0$
$\dfrac{\mathrm dy}{\mathrm dx}(x+14y)=-(14x+y)$
$\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac{14x+y}{x+14y}$

The slope of the tangent line to the curve at $(a,b)$ is then $-\dfrac{14a+b}{a+14b}$ , so the tangent to $(1,1)$ has slope

$\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{x=1,y=1}=-\dfrac{15}{15}=-1$

The point-slope form of the tangent line is then

$y-1=-(x-1)\iff y=-x+2$

7 0

3 years ago

Give an example and explain why a polynomial can have fewer x-intercepts than its number of roots.

NISA [10]

Answer:
A polynomial can have fewer x-intercepts than its number of roots when a pair of complex conjugate roots exist.

Example:
Consider the 4-th degree polynomial
f(x) = x⁴ - x³ - x² - x - 2

According to the Remainder theorem
f(-1) = 1 + 1 - 1 + 1 - 2 = 0
Therefore (x + 1) is a factor.

f(2) = 16 - 8 - 4 - 2 - 2 = 0
Therefore (x - 2) is a factor.

(x+1)*(x-2) = x² - 2x + x - 2 = x² - x - 2
To find the remaining factor, perform long division.
x² + 1
-------------------------------
x²-x-2 | x⁴ - x³ - x² - x - 2
x⁴ - x³ - 2x²
-----------------------------
x² - x - 2
x² - x - 2
Therefore
f(x) = (x+1)(x-2)(x²+1)

Notice that (x² + 1) has no real factors.
However,
x² + 1 = (x + i)(x - i),
so it has a pair of conjugate zeros +i and -i.

A graph of the function confirms that there are only two real zeros (shown in red color).